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CHAPTER XI. GENERAL THEORY OF LIE DERIVATIVES

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It has been clari_ed recently that one can de_ne the generalized Lie derivative

~ L

(_;_)f of any smooth map f : M ! N with respect to a pair of vector _elds

_ on M and _ on N. Given a section s of a vector bundle E ! M and a

projectable vector _eld _ on E over a vector _eld _ on M, the second component

L_s: M ! E of the generalized Lie derivative ~ L

(_;_)s is called the Lie derivative

of s with respect to _. We _rst show how this approach generalizes the classical

cases of Lie di_erentiation. We also present a simple, but useful comparison

of the generalized Lie derivative with the absolute derivative with respect to a

general connection. Then we prove that every linear natural operator commutes

with Lie di_erentiation. We deduce a similar condition in the non linear case

as well. An operator satisfying the latter condition is said to be in_nitesimally

natural. We prove that every in_nitesimally natural operator is natural on the

category of oriented m-dimensional manifolds and orientation preserving local

di_eomorphisms.

A signi_cant advantage of our general theory is that it enables us to study

the Lie derivatives of the morphisms of _bered manifolds (our feeling is that the

morphisms of _bered manifolds are going to play an important role in di_erential

geometry). To give a deeper example we discuss the Euler operator in the higher

order variational calculus on an arbitrary _bered manifold. In the last section

we extend the classical formula for the Lie derivative with respect to the bracket

of two vector _elds to the generalized Lie derivatives.

47. The general geometric approach

47.1. Let M, N be two manifolds and f : M ! N be a map. We recall that

a vector _eld along f is a map ': M ! TN satisfying pN _ ' = f, where

pN : TN ! N is the bundle projection.

Consider further a vector _eld _ on M and a vector _eld _ on N.

De_nition. The generalized Lie derivative ~ L

(_;_)f of f : M ! N with respect

to _ and _ is the vector _eld along f de_ned by

(1) ~ L

(_;_)f : Tf _ _ 􀀀 _ _ f:

By the very de_nition, ~ L

(_;_) is the zero vector _eld along f if and only if the

vector _elds _ and _ are f-related.

Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993

47. The general geometric approach 377

47.2. De_nition 47.1 is closely related with the kinematic approach to Lie differentiation.

Using the ows Fl_

t and Fl_

t of vector _elds _ and _, we construct

a curve

(1) t 7! (Fl_

􀀀t

_f _ Fl_

t )(x)

for every x 2 M. Di_erentiating it with respect to t for t = 0 we obtain the

following

Lemma. ~ L

(_;_)f(x) is the tangent vector to the curve (1) at t = 0, i.e.

~ L

(_;_)f(x) = @

0 (Fl_

􀀀t

_f _ Fl_

t )(x):

47.3. In the greater part of di_erential geometry one meets de_nition 47.1 in

certain more speci_c situations. Consider _rst an arbitrary _bered manifold

p: Y ! M, a section s: M ! Y and a projectable vector _eld _ on Y over a

vector _eld _ on M. Then it holds Tp _ (Ts _ _ 􀀀 _ _ s) = 0M, where 0M means

the zero vector _eld on M. Hence ~L(_;_)s is a section of the vertical tangent

bundle of Y . We shall write

~ L

(_;_)s =: ~ L_s: M ! V Y

and say that ~ L_ is the generalized Lie derivative of s with respect to _. If we

have a vector bundle E ! M, then its vertical tangent bundle V E coincides

with the _bered product E _M E, see 6.11. Then the generalized Lie derivative

~ L_s has the form

~ L_s = (s;L_s)

where L_s is a section of E.

47.4. De_nition. Given a vector bundle E ! M and a projectable vector _eld

_ on E, the second component L_s: M ! E of the generalized Lie derivative

~ L_s is called the Lie derivative of s with respect to the _eld _.

If we intend to contrast the Lie derivative L_s with the generalized Lie derivative

~ L_s, we shall say that L_s is the restricted Lie derivative. Using the fact

that the second component of ~ L_s is the derivative of Fl_

􀀀t

_s _ Fl_

t for t = 0 in

the classical sense, we can express the restricted Lie derivative in the form

(1) (L_s)(x) = lim

t!0

t (Fl_

􀀀t

_s _ Fl_

􀀀s)(x):

47.5. It is useful to compare the general Lie di_erentiation with the covariant

di_erentiation with respect to a general connection 􀀀: Y ! J1Y on an arbitrary

_bered manifold p: Y ! M. For every _0 2 TxM, let 􀀀(y)(_0) be its lift to the

horizontal subspace of 􀀀 at p(y) = x. For a vector _eld _ on M, we obtain in this

way its 􀀀-lift 􀀀_, which is a projectable vector _eld on Y over _. The connection

map !􀀀 : TY ! V Y means the projection in the direction of the horizontal

Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993

378 Chapter XI. General theory of Lie derivatives

subspaces of 􀀀. The generalized covariant di_erential ~r􀀀s of a section s of Y is

de_ned as the composition of !􀀀 with Ts. This gives a linear map TxM ! Vs(x)Y

for every x 2 M, so that ~r􀀀s can be viewed as a section M ! V Y T_M, which

was introduced in another way in 17.8. The generalized covariant derivative ~r

􀀀_

of s with respect to a vector _eld _ on M is then de_ned by the evaluation

􀀀_

s := h_; ~r􀀀si : M ! V Y:

Proposition. It holds

􀀀_

s = ~ L􀀀_s:

Proof. Clearly, the value of !􀀀 at a vector _0 2 TyY can be expressed as

!􀀀(_0) = _0 􀀀 􀀀(y)(Tp(_0)). Hence ~ L􀀀_s(x) = Ts(_(x)) 􀀀 􀀀_(s(x)) coincides

with !􀀀(Ts(_(x))). _

In the case of a vector bundle E ! M, we have V E = E _ E and ~r

􀀀_

s =

(s;r􀀀_ s). The second component r􀀀_ : M ! E is called the covariant derivative

of s with respect to _, see 11.12. In such a situation the above proposition implies

(1) r􀀀_ s = L􀀀_s:

47.6. Consider further a natural bundle F : Mfm ! FM. For every vector

_eld _ on M, its ow prolongation F_ is a projectable vector _eld on FM over

_. If F is a natural vector bundle, we have V FM = FM _ FM.

De_nition. Given a natural bundle F, a vector _eld _ on a manifold M and a

section s of FM, the generalized Lie derivative

~ LF_s =: ~ L_ : M ! V FM

is called the generalized Lie derivative of s with respect to _. In the case of a

natural vector bundle F,

LF_s =: L_s: M ! FM

is called the Lie derivative of s with respect to _.

47.7. An important feature of our general approach to Lie di_erentiation is that

it enables us to study the Lie derivatives of the morphisms of _bered manifolds.

In general, consider two _bered manifolds p: Y ! M and q : Z ! M over the

same base, a base preserving morphism f : Y ! Z and a projectable vector _eld

_ or _ on Y or Z over the same vector _eld _ onM. Then Tq_(Tf__􀀀__f) = 0M,

so that the values of the generalized Lie derivative ~ L

(_;_)f lie in the vertical

tangent bundle of Z.

De_nition. If Z is a vector bundle, then the second component

(_;_)f : Y ! Z

of ~ L

(_;_)f : Y ! V Z is called the Lie derivative of f with respect to _ and _.

Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993

47. The general geometric approach 379

47.8. Having two natural bundles FM, GM and a base-preserving morphism

f : FM ! GM, we can de_ne the Lie derivative of f with respect to a vector

_eld _ on M. In the case of an arbitrary G, we write

(1) ~ L

(F_;G_)f =: ~ L_f : FM ! V GM:

If G is a natural vector bundle, we set

(2) L

(F_;G_)f =: L_f : FM ! GM:

47.9. Linear vector _elds on vector bundles. Consider a vector bundle

p : E ! M. By 6.11, Tp: TE ! TM is a vector bundle as well. A projectable

vector _eld _ on E over _ on M is called a linear vector _eld, if _ : E ! TE is a

linear morphism of E ! M into TE ! TM over the base map _ : M ! TM.

Proposition. _ is a linear vector _eld on E if and only if its ow is formed by

local linear isomorphisms of E.

Proof. Let xi, yp be some _ber coordinates on E such that yp are linear coordinates

in each _ber. By de_nition, the coordinate expression of a linear vector

_eld _ is

(1) _i(x) @

@xi + _p

q (x)yq @

@yp :

Hence the di_erential equations of the ow of _ are

dxi

dt = _i(x); dyp

dt = _p

q (x)yq:

Their solution represents the linear local isomorphisms of E by virtue of the

linearity in yp. On the other hand, if the ow of _ is linear and we di_erentiate

it with respect to t, then _ must be of the form (1). _

47.10. Let __ be another linear vector _eld on another vector bundle _E ! M

over the same vector _eld _ on the base manifold M. Using ows, we de_ne a

vector _eld _ __ on the tensor product E _E by

_ __ = @

0 (Fl_

t ) (Fl__

t ):

Proposition. _ __ is the unique linear vector _eld on E _E over _ satisfying

(1) L___(s _s) = (L_s) _s + s (L___s)

for all sections s of E and _s of _E .

Proof. If 47.9.(1) is the coordinate expression of _ and yp = sp(x) is the coordinate

expression of s, then the coordinate expression of L_s is

(2) @sp(x)

@xi _i(x) 􀀀 _p

q (x)sq(x):

Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993

380 Chapter XI. General theory of Lie derivatives

Further, let

_i(x) @

@xi + __a

b (x)zb @

@za

be the coordinate expression of __ in some linear _ber coordinates xi, za on

. If wpa are the induced coordinates on the _bers of E _E and _xi = 'i(x; t),

_yp = 'pq

(x; t)yq or _za = _'ab

(x; t)zb is the ow of _ or __, respectively, then Fl_

Fl__

_xi = 'i(x; t); _ wpa = 'pq

(x; t) _'ab

(x; t)wqb:

By di_erentiating at t = 0, we obtain

_ __ = _i(x) @

@xi + (_p

q (x)_a

b + _p

q __a

b (x))wqb @

@wpa :

Thus, if za = _sa(x) is the coordinate expression of _s, we have

L___(s _s) =

􀀀 @sp

@xi _sa + sp @_sa

@xi

_i 􀀀 _p

q sq_sa 􀀀 __a

b sp_sb:

This corresponds to the right hand side of (1). _

47.11. On the dual vector bundle E_ ! M of E, we de_ne the vector _eld __

dual to a linear vector _eld _ on E by

__ = @

0 (Fl_

􀀀t)_:

Having a vector _eld _ on M and a function f : M ! R, we can take the zero

vector _eld 0R on R and construct the generalized Lie derivative

~ L

(_;0R)f = Tf _ _ : M ! TR = R _ R:

Its second component is the usual Lie derivative L_f = _f, i.e. the derivative of

f in the direction _.

Proposition. __ is the unique linear vector _eld on E_ over _ satisfying

L_hs; _i = hL_s; _i + hs;L___i

for all sections s of E and _ of E_.

Proof. Let vp be the coordinates on E_ dual to yp. By de_nition, the coordinate

expression of __ is

_i(x) @

@xi

􀀀 _qp

(x)vq

@vp

Then we prove the above proposition by a direct evaluation quite similar to the

proof of proposition 47.10. _

47.12. A vector _eld _ on a manifold M is a section of the tangent bundle TM,

so that we have de_ned its Lie derivative L__ with respect to another vector

_eld _ on M as the second component of T_ _ _ 􀀀 T _ _ _. In 3.13 it is deduced

that L__ = [_; _]. Then 47.10 and 47.11 imply, that for the classical tensor _elds

the geometrical approach to the Lie di_erentiation coincides with the algebraic

one.

Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993

48. Commuting with natural operators 381

47.13. In the end of this section we remark that the operations with linear vector

_elds discussed here can be used to de_ne, in a short way, the corresponding

operation with linear connections on vector bundles. We recall that a linear

connection 􀀀 on a vector bundle E ! M is a section 􀀀: E ! J1E which is

a linear morphism from vector bundle E ! M into vector bundle J1E ! M.

Using local trivializations of E we _nd easily that this condition is equivalent to

the fact that the 􀀀-lift 􀀀_ of every vector _eld _ on M is a linear vector _eld on

E. By 47.9, the coordinate expression of a linear connection 􀀀 on E is

dyp = 􀀀p

qi(x)yqdxi:

Let _

􀀀

be another linear connection on a vector bundle _E ! M over the same

base with the equations

dza = _

􀀀

bi(x)zbdxi:

Using 47.10 and 47.11, we obtain immediately the following two assertions.

47.14. Proposition. There is a unique linear connection 􀀀 _

􀀀

on E _E

satisfying

(􀀀 _􀀀)(_) = (􀀀_) (_􀀀_)

for every vector _eld _ on M.

47.15. Proposition. There is a unique linear connection 􀀀_ on E_ satisfying

􀀀_(_) = (􀀀_)_ for every vector _eld _ on M.

Obviously, the equations of 􀀀 _

􀀀

are

dwpa = (􀀀p

qi(x)_a

b + _p

_􀀀

bi(x))wqbdxi

and the coordinate expression of 􀀀_ is

dvp = 􀀀􀀀q

pi(x)vqdxi:

48. Commuting with natural operators

48.1. The Lie derivative commutes with the exterior di_erential, i.e. d(LX!) =

LX(d!) for every exterior form ! and every vector _eld X, see 7.9.(5). Our

geometrical analysis of the concept of the Lie derivative leads to a general result,

which clari_es that the speci_c property of the exterior di_erential used in the

above formula is its linearity.

Proposition. Let F and G be two natural vector bundles and A: F G be a

natural linear operator. Then

(1) AM(LXs) = LX(AMs)

for every section s of FM and every vector _eld X on M.

In the special case of a linear natural transformation this is lemma 6.17.

Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993

382 Chapter XI. General theory of Lie derivatives

Proof. The explicit meaning of (1) is AM(LFXs) = LGX(AMs). By the Peetre

theorem, AM is locally a di_erential operator, so that AM commutes with limits.

Hence

AM(LFXs) = lim

t!0

􀀀

F(FlX􀀀

t) _ s _ FlXt

􀀀 AMs

= lim

t!0

G(FlX􀀀

_AMs _ FlXt

􀀀AMs

= LGX(AMs)

by linearity and naturality. _

48.2. A reasonable result of this type can be deduced even in the non linear case.

Let F and G be arbitrary natural bundles on Mfm, D: C1(FM) ! C1(GM)

be a local regular operator and s: M ! FM be a section. The generalized

Lie derivative ~ LXs is a section of V FM, so that we cannot apply D to ~ LXs.

However, we can consider the so called vertical prolongation V D: C1(V FM) !

C1(V GM) of the operator D. This can be de_ned as follows.

In general, let N ! M and N0 ! M be arbitrary _bered manifolds over the

same base and let D: C1(N) ! C1(N0) be a local regular operator. Every

local section q of V N ! M is of the form @

0 st, st 2 C1(N) and we set

(1) V Dq = V D( @

0 st) = @

0 (Dst) 2 C1(V N0):

We have to verify that this is a correct de_nition. By the nonlinear Peetre

theorem the operator D is induced by a map D : J1N ! N0. Moreover each

in_nite jet has a neighborhood in the inverse limit topology on J1N on which D

depends only on r-jets for some _nite r. Thus, there is neighborhood U of x in M

and a locally de_ned smooth map Dr : JrN ! N0 such that Dst(y) = Dr(jr

yst)

for y 2 U and for t su_ciently small. So we get

(V D)q(x) = @

0 (Dr(jrx

st)) = TDr( @

0 jrx

st) = (TDr _ _)(jrx

where _ is the canonical exchange map, and thus the de_nition does not depend

on the choice of the family st.

48.3. A local regular operator D: C1(FM) ! C1(GM) is called in_nitesimally

natural if it holds

~ LX(Ds) = V D( ~ LXs)

for all X 2 X(M), s 2 C1(FM).

Proposition. If A : F G is a natural operator, then all operators AM are

in_nitesimally natural.

Proof. By lemma 47.2, 48.2.(1) and naturality we have

V AM( ~ LFXs) = V AM

0 (F(FlX􀀀

t) _ s _ FlXt

= @

0 AM

􀀀

F(FlX􀀀

t) _ s _ FlXt

= @

􀀀

G(FlX􀀀

t) _ AMs _ FlXt

= ~ LGXAMs: _

Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993

48. Commuting with natural operators 383

48.4. Let Mf+

m be the category of oriented m-dimensional manifolds and orientation

preserving local di_eomorphisms.

Theorem. Let F and G be two bundle functors onMf+

m, M be an oriented mdimensional

manifold and let AM : C1(FM) ! C1(GM) be an in_nitesimally

natural operator. Then AM is the value of a unique natural operator A: F G

on M.

We shall prove this theorem in several steps.

48.5. Let us _x an in_nitesimally natural operator D: C1(FRm) ! C1(GRm)

and let us write S and Q for the standard _bers F0Rm and G0Rm. Since each

local operator is locally of _nite order by the nonlinear Peetre theorem, there is

the induced map D: T1

m S ! Q. Moreover, at each j1

0 s 2 T1

m S the application

of the Peetre theorem (with K = f0g) yields a smallest possible order r = _(j1

0 s)

such that for every section q with jr

0q = jr

0s we have Ds(0) = Dq(0), see 23.1.

Let us de_ne ~ Vr _ T1

m S as the subset of all jets with _(j1

0 s) _ r. Let Vr be the

interior of ~ Vr in the inverse limit topology and put Ur := _1

r (Vr) _ Trm

The Peetre theorem implies T1

m S = [rVr and so the sets Vr form an open

_ltration of T1

m S. On each Vr, the map D factors to a map Dr : Ur ! Q.

___ ____ D2

\\ \\ \\ \\ \\ \^

y w

NNX NN NQ

'1 y w '' '' '' *''

_ _ _

m S

Since there are the induced actions of the jet groups Gr+k

m on Trm

S (here k is

the order of F), we have the fundamental _eld mapping _(r) : gr+k

! X(Trm

and we write _Q for the fundamental _eld mapping on Q. There is an analogy

to 34.3.

Lemma. For all X 2 gr+k

m , jr

0s 2 Trm

S it holds

_(r)

X (jr

0s) = _(jr

0 ( ~ L􀀀Xs)):

Proof. Write _ for the action of the jet group on Trm

S. We have

_(r)

X (jr

0s) = @

0 _(exptX)(jr

0s) = @

0 jr

0 (F(FlXt

) _ s _ FlX􀀀

= _(jr

0 ( @

0 (F(FlXt

) _ s _ FlX􀀀

t))) = _(jr

0 ( ~ L􀀀Xs)): _

Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993

384 Chapter XI. General theory of Lie derivatives

48.6. Lemma. For all r 2 N and X 2 gr+k

m we have TDr _ _(r)

X = _Q

_ Dr on

Ur.

Proof. Recall that (V D)q(0) = (TDr _ _)(jr

0q) for all jr

0q 2 _􀀀1(TUr). Using

the above lemma and the in_nitesimal naturality of D we compute

(TDr _ _(r)

X )(jr

0s) = TDr(_(jr

0 ( ~ L􀀀Xs))) = V D( ~ L􀀀Xs)(0) =

= ~ L􀀀X(Ds)(0) = _Q

X(Ds(0)) = _Q

X(Dr(jr

0s)): _

48.7. Lemma. The map D: T1

m S ! Q is G1

+-equivariant.

Proof. Given a = j1

0 f 2 G1

+ and y = j1

0 s 2 T1

m S we have to show D(a:y) =

a:D(y). Each a is a composition of a jet of a linear map f and of a jet from

the kernel B1

1 of the jet projection _1

1 . If f is linear, then there are linear

maps gi, i = 1; 2; : : : ; l, lying in the image of the exponential map of G1

m such

that f = g1 _ : : : _ gl. Since T1

m S = [rVr there must be an r 2 N such that y

and all elements (j1

0 gp _ : : : _ j1

0 gl) _ y are in Vr for all p _ l. Thus, D(a:y) =

Dr(jr+k

0 f:jr

0s) = jr+k

0 f:Dr(jr

0s) = a:D(y), for Dr preserves all the fundamental

_elds.

Since the whole kernel Br

1 lies in the image of the exponential map for each

r < 1, an analogous consideration for j1

0 f 2 B1

1 completes the proof of the

lemma. _

48.8. Lemma. The natural operator A on Mf+

m which is determined by the

+-equivariant map D coincides on Rm with the operator D.

Proof. There is the associated map A: J1FRm ! GRm to the operator ARm.

Let us write A0 for its restriction (J1F)0Rm ! G0Rm and similarly for the

map D corresponding to the original operator D. Now let tx : Rm ! Rm be

the translation by x. Then the map A (and thus the operator A) is uniquely

determined by A0 since by naturality of A we have (t􀀀x)_ _ ARm _ (tx)_ = ARm.

But tx is the ow at time 1 of the constant vector _eld X. For every vector _eld

X and section s we have

~ LX((FlXt

)_s) = ~ LX(F(FlX􀀀

t) _ s _ FlXt

) = @

@t (F(FlX􀀀

t) _ s _ FlXt

)

= T(F(FlX􀀀

t)) _ ~ LXs _ FlXt

= (FlXt

)_( ~ LXs)

and so using in_nitesimal naturality, for every complete vector _eld X we compute

􀀀

(FlX􀀀

t)_(D(FlXt

)_s)

= 􀀀(FlX􀀀

t)_ ~ LX(D(FlXt

)_s) + (FlX􀀀

t)_􀀀

(V D)((FlXt

)_ ~ LXs)

= (FlX􀀀

t)_􀀀

􀀀 ~ LX(D(FlXt

)_s) + (V D)( ~ LX((FlXt

)_s))

= 0:

Thus (t􀀀x)_ _ D _ (tx)_ = D and since A0 = D0 this completes the proof. _

Lemmas 48.7 and 48.8 imply the assertion of theorem 48.4. Indeed, ifM = Rm

we get the result immediately and it follows for general M by locality of the

operators in question.

Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993

48. Commuting with natural operators 385

48.9. As we have seen, if F is a natural vector bundle, then V F is naturally

equivalent to F _ F and the second component of our general Lie derivative is

just the usual Lie derivative. Thus, the condition of the in_nitesimal naturality

becomes the usual form D_LX = LX _D if D: C1(FM) ! C1(GM) is linear.

More generally, if F is a sum F = E1 _ _ _ _ _ Ek of k natural vector bundles,

G is a natural vector bundle and D is k-linear, then we have

pr2 _ V D( ~ LX(s1; : : : ; sk)) = @

0 D

􀀀

F(FlX􀀀

t) _ (s1; : : : ; sk) _ FlXt

i=1

D(s1; : : : ;LXsi; : : : ; sk):

Hence for the k-linear operators we have

Corollary. Every natural k-linear operator A : E1 _ _ _ _ _ Ek F satis_es

(1) LXAM(s1; : : : ; sk) =

i=1 AM(s1; : : : ;LXsi; : : : ; sk)

for all s1 2 C1E1M,: : : ,sk 2 C1EkM, X 2 C1TM.

Formula (1) covers, among others, the cases of the Frolicher-Nijenhuis bracket

and the Schouten bracket discussed in 30.10 and 8.5.

48.10. The converse implication follows immediately for vector bundle functors

on Mf+

m. But we can prove more.

Let E1; : : : ;Ek be r-th order natural vector bundles corresponding to actions

_i of the jet group Gr

m on standard _bers Si, and assume that with the restricted

actions _ijG1

m the spaces Si are invariant subspaces in spaces of the

form _j(pjRm qjRm_). In particular this applies to all natural vector bundles

which are constructed from the tangent bundle. Given any natural vector

bundle F we have

Theorem. Every local regular k-linear operator

AM : C1(E1M) _ _ _ _ _ C1(EkM) ! C1(FM)

over an m-dimensional manifold M which satis_es 48.9.1 is a value of a unique

natural operator A on Mfm.

The theorem follows from the theorem 48.4 and the next lemma

Lemma. Every k-linear natural operator A : E1 _ _ _ _ _ Ek F on Mf+

extends to a unique natural operator on Mfm.

Let us remark, the proper sense of this lemma is that every operator in question

obeys the necessary commutativity properties with respect to all local di_eomorhpisms

between oriented m-manifolds and hence determines a unique natural

operator over the whole Mfm.

Proof. By the multilinear Peetre theorem A is of some _nite order `. Thus A is

determined by the associated k-linear (Gr+`

m )+-equivariant map A: T`m S1_: : :_

Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993

386 Chapter XI. General theory of Lie derivatives

T`mSk ! Q. Recall that the jet group Gr+`

m is the semidirect product of GL(m)

and the kernel Br+`

1 , while (Gr+`

m )+ is the semidirect product of the connected

component GL+(m) of the unit and the same kernel Br+`

1 . Thus, in particular

the map A: T`m S1 _ : : : _ T`m Sk ! Q is k-linear and GL+(m)-equivariant. By

the descriptions of (Gr+`

m )+ and Gr+`

m above we only have to show that any such

map is GL(m) equivariant, too. Using the standard polarization technique we

can express the map A by means of a GL+(m) invariant tensor. But looking

at the proof of the Invariant tensor theorem one concludes that the spaces of

GL+(m) invariant and of GL(m) invariant tensors coincide, so the map A is

GL(m) equivariant. _

48.11. Lie derivatives of sector forms. At the end of this section we present

an original application of proposition 48.1. This is related with the di_erentiation

of a certain kind of r-th order forms on a manifold M. The simplest case is

the `ordinary' di_erential of a classical 1-form on M. Such a 1-form ! can be

considered as a map ! : TM ! R linear on each _ber. Beside its exterior

di_erential d! : ^2TM ! R, E. Cartan and some other classical geometers used

another kind of di_erentiating ! in certain concrete geometric problems. This

was called the ordinary di_erential of ! to be contrasted from the exterior one.

We can de_ne it by constructing the tangent map T! : TTM ! TR = R _ R,

which is of the form T! = (!; _!). The second component _! : TTM ! R is

said to be the (ordinary) di_erential of ! . In an arbitrary order r we consider

the r-th iterated tangent bundle TrM = T(_ _ _ T(TM) _ _ _ ) (r times) of M.

The elements of TrM are called the r-sectors on M. Analogously to the case

r = 2, in which we have two well-known vector bundle structures pTM and

TpM on TTM over TM, on TrM there are r vector bundle structures pTr􀀀1M,

TpTr􀀀2M; : : : ; T _ _ _ TpM (r 􀀀 1 times) over Tr􀀀1M.

De_nition. A sector r-form on M is a map _ : TrM ! R linear with respect

to all r vector bundle structures on TrM over Tr􀀀1M.

A sector r-form on M at a point x is the restriction of a sector r-form an

M to the _ber (TrM)x. Denote by Tr

_M ! M the _ber bundle of all sector

r-forms at the individual points on M, so that a sector r-form on M is a section

of Tr

_M. Obviously, Tr

_M ! M has a vector bundle structure induced by the

linear combinations of R-valued maps. If f : M ! N is a local di_eomorphism

and A: (TrM)x ! R is an element of (Tr

_M)x, we de_ne (Tr

_ f)(A) = A _

(Trf􀀀1)f(x) : (TrN)f(x)

! R, where f􀀀1 is constructed locally. Since Trf is a

linear morphism for all r vector bundle structures, (Tr

_ f)(A) is an element of

(Tr

_N)f(x). Hence Tr

_ is a natural bundle. In particular, for every vector _eld X

on M and every sector r-form _ on M we have de_ned the Lie derivative

LX_ = LT r

_ X_ : M ! Tr

_M:

For every sector r-form _ : Tr ! R we can construct its tangent map T_ : TTrM

! TR = R_R, which is of the form (_; __). Since the tangent functor preserves

vector bundle structures,

__ : Tr+1M ! R

Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993

49. Lie derivatives of morphisms of _bered manifolds 387

is linear with respect to all r+1 vector bundle structures on Tr+1M over TrM,

so that this is a sector (r + 1)-form on M.

48.12. De_nition. The operator _ : C1Tr

_M ! C1Tr+1

_ M will be called the

di_erential of sector forms.

By de_nition, _ is a natural operator. Obviously, _ is a linear operator as

well. Applying proposition 48.1, we obtain

48.13. Corollary. _ commutes with the Lie di_erentiation, i.e.

_(LX_) = LX(__)

for every sector r-form _ and every vector _eld X.

49. Lie derivatives of morphisms of _bered manifolds

We are going to show a deeper application of the geometrical approach to

Lie di_erentiation in the higher order variational calculus in _bered manifolds.

For the sake of simplicity we restrict ourselves to the geometrical aspects of the

problem.

49.1. By an r-th order Lagrangian on a _bered manifold p: Y ! M we mean

a base-preserving morphism

_: JrY ! _mT_M; m = dimM:

For every section s: M ! Y , we obtain the induced m-form _ _ jrs on M.

We underline that from the geometrical point of view the Lagrangian is not a

function on JrY , since m-forms (and not functions) are the proper geometric

objects for integration on X. If xi, yp are local _ber coordinates on Y , the induced

coordinates on JrY are xi, yp_ for all multi indices j_j _ r. The coordinate

expression of _ is

L(xi; yp_)dxi ^ _ _ _ ^ dxm

but such a decomposition of _ into a function on JrY and a volume element on

M has no geometric meaning.

If _ is a projectable vector _eld on Y over _ on M, we can construct, similarly

to 47.8.(2), the Lie derivative L__ of _ with respect to _

L__ := L

(Jr_;_mT __)_: JrY ! _mT_M

which coincides with the classical variation of _ with respect to _.

49.2. The geometrical form of the Euler equations for the extremals of _ is

the so-called Euler morphism E(_) : J2rY ! V _Y _mT_M. Its geometric

de_nition is based on a suitable decomposition of L__. Here it is useful to

introduce an appropriate geometric operation.

Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993

388 Chapter XI. General theory of Lie derivatives

De_nition. Given a base-preserving morphism ': JqY ! _kT_M, its formal

exterior di_erential D': Jq+1Y ! _k+1T_M is de_ned by

D'(jq+1

x s) = d(' _ jqs)(x)

for every local section s of Y , where d means the exterior di_erential at x 2 M

of the local exterior k-form ' _ jqs on M.

If f : JqY ! R is a function, we have a coordinate decomposition

Df = (Dif)dxi

where Dif = @f

@xi +

j_j_q

@yp

_+i : Jq+1Y ! R is the so called formal (or total)

derivative of f, provided _+i means the multi index arising from _ by increasing

its i-th component by 1. If the coordinate expression of ' is ai1:::ikdxi1^_ _ _^dxik ,

then

D' = Diai1:::ikdxi ^ dxi1 ^ _ _ _ ^ dxik :

To determine the Euler morphism, it su_ces to discuss the variation L__ with

respect to the vertical vector _elds. If _p(x; y) @

@yp is the coordinate expression

of such a vector _eld, then the coordinate expression of J r_ is

j_j_r

(D__p) @

@yp

where D_ means the iterated formal derivative with respect to the multi index

_. In the following assertion we do not indicate explicitly the pullback of L__

to J2rY .

49.3. Proposition. For every r-th order Lagrangian _: JrY ! _mT_M,

there exists a morphism K(_) : J2r􀀀1Y ! V _Jr􀀀1Y _m􀀀1T_M and a unique

morphism E(_) : J2rY ! V _Y _mT_M satisfying

(1) L__ = D

􀀀

hJ r��1_;K(_)i

+ h_;E(_)i

for every vertical vector _eld _ on Y .

Proof. Write ! = dx1 ^ _ _ _ ^ dxm, !i = i @

@xi

!, K(_) =

j_j_r􀀀1 k_i

p dyp_

!i,

E(_) = Epdyp !. Since L__ = T_ _ J r_, the coordinate expression of L__ is

(2)

j_j_r

@yp

D__p:

Comparing the coe_cients of the individual expressions D__p in (1), we _nd the

following relations

(3)

Lj1:::jr

p = K(j1:::jr)

...

Lj1:::jq

p = DiKj1:::jqi

p + K(j1:::jq)

...

p = DiKji

p + Kip

Lp = DiKip

+ Ep

Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993

49. Lie derivatives of morphisms of _bered manifolds 389

where Lj1:::jq

p = _!

@yp

and Kj1:::jqi

p = _!

q! k_i

p , provided _ is the multi index

corresponding to j1 : : : jq, j_j = q. Evaluating Ep by a backward procedure, we

_nd

(4) Ep =

j_j_r

(􀀀1)j_jD_

@yp

for any K's, so that the Euler morphism is uniquely determined. The quantities

Kj1:::jqi

p , which are not symmetric in the last two superscripts, are not uniquely

determined by virtue of the symmetrizations in (3). Nevertheless, the global

existence of a K(_) can be deduced by a recurrent construction of some sections

of certain a_ne bundles. This procedure is straightforward, but rather technical.

The reader is referred to [Kol_a_r, 84b] _

We remark that one can prove easily by proposition 49.3 that a section s of

Y is an extremal of _ if and only if E(_) _ j2rs = 0.

49.4. The construction of the Euler morphism can be viewed as an operator

transforming every base-preserving morphism _: JrY ! _mT_M into a basepreserving

morphism E(_) : J2rY ! V _Y _mT_M. Analogously to L__, the

Lie derivative of E(_) with respect to a projectable vector _eld _ on Y over _

on M is de_ned by

L_E(_) := L

(J2r_;V___mT __)E(_):

An important question is whether the Euler operator commutes with Lie

di_erentiation. From the uniqueness assertion in proposition 49.3 it follows that

E is a natural operator and from 49.3.(4) we see that E is a linear operator.

49.5. We _rst deduce a general result of such a type. Consider two natural

bundles over m-manifolds F and H, a natural surjective submersion q : H ! F

and two natural vector bundles over m-manifolds G and K.

Proposition. Every linear natural operator A: (F;G) (H;K) satis_es

L_(Af) = A(L_f)

for every base-preserving morphism f : FM ! GM and every vector _eld _ on

Proof. By 47.8.(2) and an analogy of 47.4.(1), we have

L_f = lim

t!0

G(Fl_

􀀀t) _ f _ F(Fl_

t ) 􀀀 f

Since A commutes with limits by 19.9, we obtain by linearity and naturality

AL_(f) = lim

t!0

K(Fl_

􀀀t) _ Af _ H(Fl_

t ) 􀀀 Af

= L_(Af) _

Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993

390 Chapter XI. General theory of Lie derivatives

49.6. Our original problem on the Euler morphism can be discussed in the same

way as in the proof of proposition 49.5, but the functors in question are de_ned

on the local isomorphisms of _bered manifolds. Hence the answer to our problem

is a_rmative.

Proposition. It holds

L_E(_) = E(L__)

for every r-th order Lagrangian _ and every projectable vector _eld _ on Y .

49.7. A projectable vector _eld _ on Y is said to be a generalized in_nitesimal

automorphism of an r-th order Lagrangian _, if L_E(_) = 0. By proposition

49.6 we obtain immediately the following interesting assertion.

Corollary. Higher order Noether-Bessel-Hagen theorem. A projectable

vector _eld _ is a generalized in_nitesimal automorphism of an r-th order Lagrangian

_ if and only if E(L__) = 0.

49.8. An in_nitesimal automorphism of _ means a projectable vector _eld _

satisfying L__ = 0. In particular, corollary 49.7 and 49.3.(4) imply that every

in_nitesimal automorphism is a generalized in_nitesimal automorphism.

50. The general bracket formula

50.1. The generalized Lie derivative of a section s of an arbitrary _bered manifold

Y ! M with respect to a projectable vector _eld _ on Y over _ on M is

a section ~ L_s: M ! V Y . If __ is another projectable vector _eld on Y over __

on M, a general problem is whether there exists a reasonable formula for the

generalized Lie derivative ~ L

[_;__]s of s with respect to the bracket [_; __]. Since

L_s is not a section of Y , we cannot construct the generalized Lie derivative of

~ L_s with respect to __. However, in the case of a vector bundle E ! M we have

de_ned L__L_s: M ! E.

Proposition. If _ and __ are two linear vector _elds on a vector bundle E ! M,

then

(1) L

[_;__]s = L_L__s 􀀀 L__L_s

for every section s of E.

At this moment, the reader can prove this by direct evaluation using 47.10.(2).

But we shall give a conceptual proof resulting from more general considerations

in 50.5. By direct evaluation, the reader can also verify that the above proposition

does not hold for arbitrary projectable vector _elds _ and __ on E. However,

if FM is a natural vector bundle, then F_ is a linear vector _eld on FM for

every vector _eld _ on M, so that we have

Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993

50. The general bracket formula 391

Corollary. If FM is a natural vector bundle, then

[_;__]s = L_L__s 􀀀 L__

L_s

for every section s of FM and every vector _elds _, __ on M.

This result covers the classical cases of Lie di_erentiation.

50.2. We are going to discuss the most general situation. Let M, N be two

manifolds, f : M ! N be a map, _, __ be two vector _elds on M and _, __ be two

vector _elds on N. Our problem is to _nd a reasonable expression for

(1) ~ L

([_;__];[_;__])f : M ! TN:

Since ~ L

(_;_)f is a map of M into TN, we cannot construct its Lie derivative

with respect to the pair (__; __), since __ is a vector _eld on N and not on TN.

However, if we replace __ by its ow prolongation T __, we have de_ned

(2) ~ L

(__;T __)

~ L

(_;_)f : M ! TTN:

On the other hand, we can construct

(3) ~ L

(_;T _)

~ L

(__;__)f : M ! TTN:

Now we need an operation transforming certain special pairs of the elements

of the second tangent bundle TTQ of any manifold Q into the elements of TQ.

Consider A,B 2 TTzQ satisfying

(4) _TQ(A) = T_Q(B) and T_Q(A) = _TQ(B):

Since the canonical involution _: TTQ ! TTQ exchanges both projections, we

have _TQ(A) = _TQ(_B), T_Q(A) = T_Q(_B). Hence A and _B are in the

same _ber of TTQ with respect to projection _TQ and their di_erence A 􀀀 _B

satis_es T_Q(A􀀀_B) = 0. This implies that A􀀀_B is a tangent vector to the

_ber TzQ of TQ and such a vector can be identi_ed with an element of TzQ,

which will be denoted by A _ B.

50.3. De_nition. A _ B 2 TQ is called the strong di_erence of A, B 2 TTQ

satisfying 50.2.(4).

In the case Q = Rm we have TTRm = Rm _ Rm _ Rm _ Rm. If A =

(x; a; b; c) 2 TTRm, then B satisfying 50.2.(4) is of the form B = (x; b; a; d) and

one _nds easily

(1) A _ B = (x; c 􀀀 d)

From the geometrical de_nition of the strong di_erence it follows directly

Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993

392 Chapter XI. General theory of Lie derivatives

Lemma. If A, B 2 TTQ satisfy 50.2.(4) and f : Q ! P is any map, then

TTf(A), TTf(B) 2 TTP satisfy the condition of the same type and it holds

TTf(A) _ TTf(B) = Tf(A _ B) 2 TP:

50.4. We are going to deduce the bracket formula for generalized Lie derivatives.

First we recall that lemma 6.13 reads

(1) [_; __] = T __ _ _ _ T_ _ __

for every two vector _elds on the same manifold.

The maps 50.2.(2) and 50.2.(3) satisfy the condition for the existence of

the strong di_erence. Indeed, we have _TN _ ~ L

(__;T __)

~ L

(_;_)f = ~ L

(_;_)f since

any generalized Lie derivative of ~ L

(_;_)f is a vector _eld along ~ L

(_;_)f. On

the other hand, T_N _ ( ~ L

(__;T __)

~ L

(_;_)f) = T_N

􀀀 @

0 T(Fl__

􀀀t) _ ~ L

(_;_)f _ Fl__

0 (Fl__

􀀀t

_f _ Fl__

t ) = ~ L(__; __)f.

Proposition. It holds

(2) ~ L

([_;__];[_;__])f = ~ L

(_;T _)

~ L

(__;__)f _ ~ L

(__;T __)

~ L

(_;_)f

Proof. We _rst recall that the ow prolongation of _ satis_es T _ = _ _ T_. By

47.1.(1) we obtain ~ L

(__;T __)

~ L

(_;_)f = T(Tf _ _ 􀀀_ _ f) _ __ 􀀀T __ _ (Tf _ _ 􀀀_ _ f) =

TTf _ T_ _ __ 􀀀 T_ _ Tf _ __ 􀀀 _ _ T __ _ Tf _ _ + _ _ T __ _ _ _ f as well as a similar

expression for ~ L

(_;T _)

~ L

(__;__)f. Using (1) we deduce that the right hand side of

(2) is equal to Tf _(T_ _ ___T __ __)􀀀(T_ _ ___T __ __)f = Tf _[_; __]􀀀[_; __]_f. _

50.5. In the special case of a section s: M ! Y of a _bered manifold Y ! M

and of two projectable vector _elds _ and __ on Y , 50.4.(2) is specialized to

(1) ~ L

[_;__]s = ~ LV_ ~ L__s _ ~ LV__ ~ L_s

where V_ or V __ is the restriction of T _ or T __ to the vertical tangent bundle

V Y _ TY . Furthermore, if we have a vector bundle E ! M and a linear vector

_eld on E, then V_ is of the form V_ = _ _ _, since the tangent map of a linear

map coincides with the original map itself. Thus, if we separate the restricted Lie

derivatives in (1) in the case _ and __ are linear, we _nd L

[_;__]s = L_L__s􀀀L__L_s.

This proves proposition 50.1.

Remarks

The general concept of Lie derivative of a map f : M ! N with respect to a

pair of vector _elds on M and N was introduced by [Trautman, 72]. The operations

with linear vector _elds from the second half of section 47 were described

Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993

Remarks 393

in [Jany_ska, Kol_a_r, 82]. In the theory of multilinear natural operators, the commutativity

with the Lie di_erentiation is also used as the starting point, see

[Kirillov, 77, 80]. Proposition 48.4 was proved by [Cap, Slov_ak, 92]. According

to [Jany_ska, Modugno, to appear], there is a link between the in_nitesimally natural

operators and certain systems in the sense of [Modugno, 87a]. The concept

of a sector r-form was introduced in [White, 82].

The Lie derivatives of morphisms of _bered manifolds were studied in [Kol_a_r,

82a] in connection with the higher order variational calculus in _bered manifolds.

We remark that a further analysis of formula 49.3.(3) leads to an interesting fact

that a Lagrangian of order at least three with at least two independent variables

does not determine a unique Poincar_e-Cartan form, but a family of such forms

only, see e.g. [Kol_a_r, 84b], [Saunders, 89]. The general bracket formula from

section 50 was deduced in [Kol_a_r, 82c].

Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993

394

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